Doomsday Prediction I have a calculus problem I can't seem to figure out. Any help would be appreciated! Doomsday prediction. In $1960$, three electrical engineers at the University of Illinois published a paper in Science titled "Doomsday." Based on world population growth data from $1000$ AD to $1960$ AD, the engineers found that population growth was faster than proportional to the population size. Using the data, they modeled the growth of the population as $dP/dt = 0.4873P^2 \:\:$ $P(0) = 0.2$ where $P$ is the population size in billions and $t$ is centuries after $1000$ AD. 1) Solve this differential equation.

As you have a "nice form" differential equation here, you can use Separation of Variables and integration to solve. Note that

$$\frac{dP}{dt}=0.4873P^2$$

can be written as

$$\frac{dP}{P^2}=0.4873dt$$

Now you can integrate both sides

$$\int\frac{dP}{P^2}=\int0.4873dt$$ Thus

$$-\frac{1}{P}+C_P=0.4873t+C_t$$

To finish, note that $C_P$ and $C_t$ are arbitrary constants due to the indefinite integrals. Thus we can rewrite as

$$-\frac{1}{P}=0.4873t+C$$

which is the same as

$$P=\frac{1}{C-0.4873t}$$

for some real $C$. Now as $P(0)=0.2, C=5$ So finally we have

$$P=\frac{1}{5-0.4873t}$$

As the denominator approaches $0$, the population approaches $\infty$. This occurs when $5-0.4873t=0$ Thus $t=10.2606197....$ and as $t$ is the number of centuries after $1000$ AD, we can expect an infinite population in $10$ years from now.